Number of points of curves over finite fields in some relative situations from an euclidean point of vue

TL;DR

This paper investigates the number of rational points on smooth projective curves over finite fields using Euclidean geometry, deriving relative Weil bounds through inequalities on geometric subspaces.

Contribution

It introduces new relative Weil bounds for rational points on curves, based on Euclidean geometric methods and inequalities involving the Frobenius graph.

Findings

Derived relative Weil bounds for rational points

Applied Euclidean inequalities to geometric subspaces

Extended previous results to new relative situations

Abstract

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.